This question is related to this question about the Machina paradox and about the expected utility model. In this question, I'd like to know a little more about various or even competing ways of specifying utility and decision making. It'd be nice if we could formulate a list of different formulations. To start, I suppose there is the Von Neumann–Morgenstern expected utility model, Savage's subjective expected utility model, or Prospect theory.

4 Answers
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This won't get at individual choice, but how about evolutionary approaches? Perhaps this isn't what you are looking for, but one way to model decisions is to wander from the rational paradigm entirely. All changes in behavior are driven by natural selection, and so an equilibrium is based on stability.

In a symmetric normal form game, an evolutionarily
stable strategy is a (possibly mixed) strategy with the following property: a population in
which all members play this strategy is resistant to invasion by a small group of mutants
who play an alternative mixed strategy.

Edit: Wanted to add some more exotic models, inspired by @CompEcon. These are interesting models for temptation, inconsistency, etc because you can get at internal conflict by seeing a DM as the unification of multiple parts.

We provide a user's guide to 'exotic' preferences: nonlinear time aggregators, departures from expected utility, preferences over time with known and unknown probabilities, risk-sensitive and robust control, 'hyperbolic' discounting, and preferences over sets ('temptations'). We apply each to a number of classic problems in macroeconomics and finance, including consumption and saving, portfolio choice, asset pricing, and Pareto optimal allocations.

The paper itself does a great job of overviewing many options for "non-standard" preferences and utility. They introduce a preference, outline key features, and then apply them in a number of classic settings to give you a feel for what is going on.

If you are interested in using non-traditional preferences, you should certainly read this. I can't give extensive insight further than that -- I currently do nearly all my work with more traditional CRRA preferences.

The traditional graduate micro theory education develops the neoclassic a model of consumer choice along the two parallel lines:

Observed choice based on the weak axiom of revealed preference.

The preference relation approach, based on the rational (i.e. transitive and complete) preferences $\succsim$. Rational preferences plus continuity are sufficient for the existence of a utility function.

We know that the two offer very similar foundations for a model of consumer choice, but that they are not quite equivalent. For one, any demand system generated by a rational preference relation will always have a symmetric substitution matrix, whereas the substitution matrix for a demand system based on WARP may be asymmetric. In that sense, the WARP framework is less restrictive. The usual goto discussion of this can be found in Chapters 1-3 of Mas-Colell et al.'s "Microeconomic Theory".

Weak order relations, they are sets $\succ \in X\times X$ such that $(a,b)\in\succ $ is denoted $a\succ b$ . This is the usual language in most of decision theory.

Utility functions or value functions, they have the following structure $u:X\times A \mapsto \mathbb{R} $ where the value depends on the alternative and the menu. Then the behavior is described in terms of maximizing such object subject to some constraint than sometimes is just the menu $C(A)=argmax_{x\in A}u(x,A)$.